Descriptive summaries

Data was visually summarised with plots by age-gender-BMI groups. Medians and confidence, weighted to adjust for sampling strategies, summarised are used as inputs for the DiSCAO and presented in the Appendix.

Estimating % Fat mass

Body composition is a required input into the energy regulation equations presented by Hall et al 2013 and used in Chapter 4 Section 4.2.2.3.

\[ \Delta FFM_{i,j,k}=\frac{pEnS_{i.j.k}+G_{j,k}}{\hat{\rho_{FFM, i,j,k}}}\] and

\[ \Delta FM_{i,j,k}=\frac{(1-p_{i,j,k})EnS_{i.j.k}-G_{j,k}}{\rho_{FM}}\] Such that \(\hat{\rho_{FFM, i,j,k}}\) and \(p_{i,j,k}\) rely on estimates of body composition.

\[ \hat{\rho_{FFM, i,j,k}} = (837+4.3 \times FFM_{i,j,k}) \times 4.18\]

With the p-ratio defined by Forbes 1987. \[ p_{i,j,k} = \frac{C_{i,j,k}}{C_{i,j,k}+FM_{i.j.k}}\]

and where,

\[C_{i,j,k}=10.4 \times\frac{\hat{\rho_{FFM, i,j,k}}}{\rho_{FM}}\]

\(i=\{ \text{BMI categoried} \}\)

\(j=\{ \text{Age groups} \}\)

\(k=\{ \text{Gender} \}\)

these equations were constructed using linear regression assuming normal distribution errors on a relatively small sample (n=416). These equations will also need to be applied to the main LSAC waves to give FFM% estimates for the required subgroups Linear regression was not appropriate for extrapolation.

Beta regression was used to develop equations to predict the %FFM for the “Checkpoint” data collection. These equations are then applied to the general LSAC data to estimate the FFM% for all sub-groups needed for the model.

A step-wise algorithm was used to sequentially test model terms ranging including combinations of gender, height, weight, and 2-way and 3-way interactions. Additionally, polynomial terms for height and weight were considered up to the 3rd order (height^3), this gives the model freedom to choose non-linear relationships.

The predicted %FFM was estimated using the Coretes-Castell equations and the beta regression to assess the appropriateness of each method.

Descriptive summaries

Individual level BMI

These histograms of BMI over time (Figure 2) and gender-age-groups show the age sampling of the dual cohort of LSAC. The shift in shape is a key underlying dynamics that reflects changes in the prevalence of obesity in Australian children.

Figure 2: Comparison of BMI distribution over time and by gender and age groups

Body weight

Figure 3 shows the reported body weight over age, by gender and BMI categories. In general body weight increases with age, overweight and obese body weights increase at a faster rate and males have a higher body weight than females. It should be noted that the data is presented as cross-sectional when there are repeat measures.

Figure 3: Combined wave, Body weight by age, gender and BMI category

Figure 4 plots reported body weight from the national health survey (NHS) for all ages.

Figure 4: 2007 National Health Survey: Body weight by age, gender and BMI category

Height

Figure 5 plots the age-gender-BMI hieghts in the LSAC data.

Figure 5: Combined wave, Height by age, gender and BMI category

Figure 6: 2007 National Health Survey: Height by age, gender and BMI category

LSAC CheckPoint data

Figures 7 and 8 are 3D plots of the LSAC CheckPoint data for males and females. These plots can be examined by spinning the plot areas to look at the relationships between body weight, height and %fat-mass.

Beta-Regression model

The beta-regression coefficients were fitted for the health CheckPoint data using weight and height. The resulting equations are used to predict the % fat mass for each gender and are applied to the whole LSAC data. The following sections examine model performance.

Males

\[ FM_{Males} = \text{Exp}(-5.236806 - \text{weight}^{2}\times 1.057483e^{-3} - \text{height}^2 \times 2.921958e^{-4} \\ + \text{weight} \times 1.319295e^{-1} + \text{height} \times 4.529665e^{-2} + \text{weight}^2 \times \text{height}^2 \times 1.690558e^{-8} - \text{weight} \times \text{height} \times 1.492729e^{-4})\]

Females

\[ FM_{females} = \text{Exp}(-5.236806 - \text{weight}^{2}\times 1.057483e^{-3} - \text{height}^2 \times 2.921958e^{-4} + \text{weight} \times 1.319295e^{-1} \\ + \text{height} \times 4.529665e^{-2} + \text{weight}^2 \times \text{height}^2 \times 1.690558e^{-8} - \text{weight} \times \text{height} \times 1.492729e^{-4} \\ + 7.542881e^{-2} - \text{weight}^{2}\times 2.553074e^{-4} + \text{height}^2 \times 7.785382e^{-6} + \text{weight}^2 \times \text{height}^2 \times 7.933846e^{-9} )\]

Residuals

Beta regression The fitted vs residual plot (Figure 8) shows a generally good fit with a slight under estimate in higher %fat percentage and overestimates in low %fat percentage. A good fit would be seen when the red loess line follows the dashed linear (x=y) line.

Figure 8: Beta regression residuals against fitted obseravtions

The distribution of the beta regression residuals are shown in Figure 9.

Figure 9: Distribution beta regression residuals

Bland-Altman Plot

The Bland-Altman plot, shows the difference between observed and predicted against observed fat percentage in the CheckPoint data.

Figure 10: Bland-Altman plots residuls vs observed

Extrapolation

The beta regression model was used to predict the remaining LSAC data. Figure 11 shows the age-gender-BMI curves for predicted % fat mass.

Figure 11: Extrapolation of Beta regression model onto LSAC data

Beta Predictions againts observed LSAC wave 4-8 data

As mentioned in the methods, the main wave of the LSAC data has %fat mass collected between waves 4-8. Figure 12 shows the fit of the Beta regression predictions against the wave 4-8 data collected. The plot has coloured points by “Data quality”, observed fat-mass from waves 4-8 with fat-mass = 0 or greater than 80% was considered to be possibly an error.

The plot shows that the predicted estimates follow the linear line with deviations in the upper and lower ends.

Figure 12: Comparison of beta regression predictions against observed LSAC data

Comparison to published % Fat mass data

Below are estimated % fat mass curves using the National Health and Nutrition Examination Survey (NHANES) IV (Laurson, Eisenmann and Welk, 2011,https://www.cooperinstitute.org/vault/2440/web/files/787.pdf ). we can see that the plots have a close resemblance to the predicted beta regression estimates.

The predictions generally follow the shape of NHANES data (Figure 13 vs Figure 14), over similar age ranges.

Figure 13: % fat mass curves using National Health and Nutrition Examination Survey (NHANES)

Figure 14: Quantile estimates of beta regression predictions

Quantile output tables

Laurson, Eisenmann and Welk: NHANES LMS Estimates

Similarly, NHANES LMS Estimates percentiles shows some agreement between published and predicted % fat mass.

Figure 15: Quantile estimates of % fat mass curves using National Health and Nutrition Examination Survey (NHANES)

From Beta Regression

gender	age	p_2	p_5	p_10	p_25	p_50	p_75	p_85	p_90	p_95	p_98
Male	2	10.38	10.9	11.47	12.34	13.51	14.79	15.56	16.14	17.1	18.53
Male	3	10.68	11.4	11.91	12.75	13.92	15.43	16.28	17.09	18.37	19.93
Male	4	11.71	12.39	13.03	14.07	15.42	16.98	18.06	19	20.9	24.45
Male	5	11.56	12.38	13.11	14.35	15.72	17.48	18.81	20.05	22.74	27.19
Male	6	12.18	12.95	13.66	14.88	16.46	18.73	20.55	22.28	25.81	31.17
Male	7	11.91	12.84	13.49	14.83	16.53	18.88	20.75	23.53	28.45	35.85
Male	8	12.25	13.15	13.89	15.53	17.72	21.32	24.97	27.66	33.73	40.69
Male	9	11.95	12.99	13.88	15.54	18.33	22.69	26.63	30.03	37.32	44.18
Male	10	12.31	13.29	14.26	16.31	19.52	25.52	29.78	33.24	39.33	46.28
Male	11	11.91	13.34	14.04	16.13	19.74	27.08	30.91	33.39	38.01	45.62
Male	12	11.14	12.37	13.46	15.74	19.16	25.4	30.4	33.69	38.67	43.75
Male	13	10.49	11.7	13.16	15.32	18.98	25.3	30.14	32.99	36.31	40.27
Male	14	10.35	11.39	12.73	14.87	18.68	24.78	29	31.08	34.58	38.16
Male	15	9.78	11.22	12.6	15.07	18.85	25.17	28.81	31.5	34.22	36.24
Male	16	10.01	11.67	12.92	15.48	19.19	25.26	28.31	30.56	32.2	35.62
Male	17	9.98	11.33	13.02	16.07	20.86	26.79	29.8	31.39	33.3	35.84
Female	2	11.24	11.8	12.33	13.2	14.31	15.67	16.49	17.23	18.31	19.86
Female	3	11.35	12.03	12.56	13.5	14.63	16.2	17.11	17.8	18.79	20.77
Female	4	12.84	13.57	14.21	15.3	16.73	18.52	19.76	20.98	23.16	26
Female	5	12.93	13.78	14.32	15.61	17.06	19.06	20.62	21.59	23.84	27.49
Female	6	13.66	14.46	15.2	16.55	18.31	20.68	22.42	24.2	27.99	32.81
Female	7	13.8	14.46	15.26	16.74	18.64	21.47	23.82	25.84	30.17	35.18
Female	8	13.83	15.03	15.88	17.7	20.18	24.24	27.21	29.71	34.42	39.25
Female	9	13.51	14.7	15.64	17.57	20.48	25.12	28.75	31.28	35.06	41.75
Female	10	13.34	14.74	15.82	18.26	21.65	27.06	31.96	34.6	38.91	42.15
Female	11	13.55	14.72	16.2	18.58	22.39	28.46	31.95	34.21	38.43	41.29
Female	12	13.22	14.9	16.46	19.44	23.74	29.43	33.26	35.87	39.26	41.03
Female	13	13.79	15.38	16.82	20.18	24.61	30.74	34.18	37.05	39.74	40.64
Female	14	13.95	16.34	18.36	21.93	26.15	31.96	35.97	38.37	39.83	41.14
Female	15	15.71	16.71	19.41	23.08	27.73	33.09	35.99	37.83	40.02	41.29
Female	16	15.5	17.51	19.77	23.87	28.73	34.75	38.19	39.29	40.58	41.36
Female	17	16.57	18.34	19.77	24.07	29.14	35.21	38.11	39.69	40.46	41.56

Weight

Heights

% Fat-mass

## R version 4.3.2 (2023-10-31)
## Platform: x86_64-pc-linux-gnu (64-bit)
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